Bayesian Anchored Latent Learning for Estimating State Trajectories

Dynamic systems often evolve faster than reliable measurements can be collected. More frequent indirect observations may fill these gaps, but their noise and differing scales complicate estimation of the underlying state. We developed Bayesian Anchored Latent Learning (BALL) to combine sparse reference measurements with denser records on an interpretable scale. A bidirectional transformer teacher reconstructs trajectories from complete training histories, and a forward-only transformer student learns from those reconstructions and observed measurements. Across 30 simulations, BALL reduced composite trajectory error by 7% relative to the same transformer trained directly from measurements, achieving lower error across all datasets. In an applied example using psychiatric records from 1,609 patients, BALL improved the prediction of withheld questionnaire scores compared with direct transformer training at 21- and 28-day measurement schedules. Requests guided by estimated uncertainty further improved prediction. BALL connects retrospective learning with prospective state estimation and provides a framework for directing additional measurement.